# What's the next number in this number sequence?

Find the next number in this sequence:

2, 4, 8, 13, 20, 29, 41, ?

EXTENSION: Find the number for the nth term in the sequence (in terms of n).

Hint:

Fibonacci+Triangle

• mathworld.wolfram.com/TribonacciNumber.html
– Duck
Commented Sep 9, 2018 at 2:33
• The hint has nothing to do with that. Commented Sep 9, 2018 at 2:34
• Ok, just checking
– Duck
Commented Sep 9, 2018 at 2:35
• And I changed it to make it more obvious. Commented Sep 9, 2018 at 2:37
• OEIS found this but I don't think that is what you had in mind Commented Sep 9, 2018 at 2:45

57

Solution for the extension: (Thanks @PerpetualJ for MathJax)

$$Fi. Δ.$$ $$1 + 1 = 2$$ $$1 + 3 = 4$$ $$2 + 6 = 8$$ $$3 + 10 = 13$$ $$5 + 15 = 20$$ $$8 + 21 = 29$$ $$13 + 28 = 41$$ $$21 + 36 = 57 \leftarrow (Next Term)$$

Round off to the nearest integer.
$$Fi(n) = round\biggl(\frac{(-\frac{1}{\phi})^n}{\sqrt{5}}\biggr)$$
$$\Delta(n) = \frac{n(n + 1)}{2}$$
$$\int(n) = Fi(n) + \Delta(n) = round\biggl(\frac{(-\frac{1}{\phi})^n}{\sqrt{5}}\biggr) + \frac{n(n + 1)}{2}$$

• Yes, good job!! Commented Sep 9, 2018 at 7:00

The next number is

$34$

The formula is

$2^n - \dfrac{(n-1)(n-2)(n-3)}2$

• I'm thinking of a different formula and I realized this worked as well so I'll edit the question. Commented Sep 9, 2018 at 2:25